We intend to continue the work done under Grant GM18770, on the theory of probability estimation, one of the most important applications of which will eventually be to medical diagnosis. The problems are different for continuous and discontinuous distributions though some variables are "mixed" in the medical application. For both classes of problems one can use the method of penalized likelihood which has both a Bayesian and a non-Bayesian interpretation. The Bayesian interpretation enables one to say how much beteer one putative density curve is then another one as measured by the technical concept of weight of evidence, and thus leads to an evaluation also of bumps or clusters. For continuous distributions we have developed this method in detail for univariate problems where the probability density function is not assumed to belong to a parameterized family. We hope to extend this work to bivariate problems for which we have some real medical data. For contingency tables and multidimensional contingency tables the "penalty" we subtract from the log-likelihood is proportional to negative entropy. We hope to apply the method of penalized likelihood also to parametric problems. Related to probability estimation is the Bayes/non-Bayes compromise which is associated with the heirarchical Bayes method. These have been applied strinkingly, by using mixed Dirichlet priors, to multinomial distributions and to contingency tables, as well as to continuous univariate problems, and we would like to make multivariate extensions. We also hope to compare our methods with those developed elsewhere.